A Characterization of Optimal Prefix Codes

TL;DR

The paper provides a complete characterization of optimal binary prefix codes for finite sources, showing that optimality is equivalent to completeness and strong monotonicity. Strong monotonicity is defined via Kraft sums: for any $A,B\subseteq S$ with $K_C(A)=2^{-i}>2^{-j}=K_C(B)$, we have $P(A)\ge P(B)$. The main result extends the Huffman-sibling-property view by showing optimality does not require the full sibling property, but can be achieved under completeness and strong monotonicity. These results clarify the structure of optimal codes and can influence code design in data compression and related optimization problems.

Abstract

A property of prefix codes called strong monotonicity is introduced, and it is proven that for a given source, a prefix code is optimal if and only if it is complete and strongly monotone.

Figures (3)

• Figure 1: Logical implications of prefix code properties for a given source. The red arrows indicate new results presented in this paper.
• Figure 2: A code tree illustrating monotonicity without strong monotonicity.
• Figure 3: An non-complete prefix code tree illustrating strong monotonicity without optimality.

Theorems & Definitions (12)

• Theorem 1.1
• Lemma 2.1: Huffman Huffman-1952
• Lemma 2.2: Kraft Inequality converse Cover-Thomas-book-2006
• Definition 2.3
• Lemma 2.4: Gallager Gallager-IT-1978
• Definition 3.1
• Example 3.2: complete & monotone $\mathrel{{\ooalign{\hidewidth$ $\hidewidth\cr$$}}}$ optimal
• Example 3.3: strongly monotone $\mathrel{{\ooalign{\hidewidth$ $\hidewidth\cr$$}}}$ optimal
• Lemma 3.4
• Lemma 3.5